Notice that \(\tilde{\psi }\)  is given by applying the involution \({1\over z} \in \mathbb {P}\mathrm {SL}(2, \)  and a permutation (which permutes the first and the second last coordinate) to \(\tilde{\phi }\) . Since \(\mathcal {M}_2\)  is defined as the quotient of \(\mathcal {C}_{\infty }\)  by the product group \(\mathbb {P}\mathrm {SL}(2,  \times S_6\) , it follows that \(\pi _{\mathcal {M}_2}(\tilde{\phi }(\tilde{\mathchoice{\chi }{}{}{}}{\chi }\) )) M2((\(\chi \)\(\chi \) 0))\( and by the commutativity of (\ref {CD}) we know that \) M2((\(\chi \)\(\chi \) ))= (\(\chi \)\(\chi \) )\(.\)
Proposition 3.4 
We compute that the pull back \(\tilde{\psi }^*(\omega ^{\scriptscriptstyle (}k\) )\(\) )\( is projectively equivalent to the \) End(Sk(F2))\( valued 1-form\begin{equation}\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{equation}\)\(\chi \)\(\chi \)  := k 1 i 5 Ai  dt t - i

where \(A_i = \widehat{\Omega }^{a,b} + \widehat{\Omega }^{c,d}\)  for \(1 \leqslant a<b,c<d \leqslant 5\)  such that \([a+b]=[c+d] = [i]\)  and \([x] := x \pmod {5}\) .

Remark 3.5 For any \(1 \leqslant i \leqslant 5\) , there exists only one solution for \(A_i\)  with the given constraints. From the expression it is obvious that the 1-form () has no poles at \(-\mu _5\)  and extends as a holomorphic 1-form to \(\tilde{\mathchoice{\chi }{}{}{}}{\chi }\)\(, which justifies the subscript in (\ref {psipullback}).\)
Let \(D^0_6 \subset \mathcal {C}_{\infty }\)  be
\(D^0_6:=\lbrace  (z_1, \dots , z_6) \in \mathcal {C}| z_6 = 0\rbrace .\) 
Then \(\tilde{\psi }(\tilde{\mathchoice{\chi }{}{}{}}{\chi }\) 0) D06\( and{\begin{@align*}{1}{-1}\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{@align*}}_{|D_6^0} &= {\hspace{-5.0pt}k}\bigg ( \sum _{1 \leqslant i < j \leqslant 5} \widehat{\Omega }^{i,j} {dz_i - dz_j \over z_i - z_j} + \sum _{i=1}^5 \widehat{\Omega }^{i,6}{dz_i \over z_i}\bigg )\\&= {\hspace{-5.0pt}k}\bigg ( \sum _{1 \leqslant i < j \leqslant 5} \widehat{\Omega }^{i,j} d \log (z_i - z_j) + \sum _{i=1}^5 \widehat{\Omega }^{i,6}d \log (z_i)\bigg ).\) 
Thus

\(\tilde{\psi }^*(\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
) = k( 1 i < j 5 i,j d (1 i + -it - 1 j + -jt)
- i=15 i,6d (i + -it)).

Now
\(d \log \bigg ({1 \over \zeta ^i + \zeta ^{-i}t} - {1 \over \zeta ^j + \zeta ^{-j}t}\bigg ) = &d \log (\zeta ^i + \zeta ^{-i}t - \zeta ^j - \zeta ^{-j}t) \\&- d\log (\zeta ^i + \zeta ^{-i}t) - d\log (\zeta ^j + \zeta ^{-j})\) 

which implies that

\(&\tilde{\psi }^*(\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
) =k ( 1 i < j 5 i,jd (i + -it - j - -jt)
- 1 i < j 5 i,jd(i + -it)
- 1 i < j 5 i,j d(j + -jt)) -  i=15 i,6 d (i + -it)).

The last three terms can be simplified by
\(\sum _{1 \leqslant i < j \leqslant 5} \widehat{\Omega }^{i,j}d\log (\zeta ^i + \zeta ^{-i}t)+\sum _{1 \leqslant i < j \leqslant 5} \widehat{\Omega }^{i,j} d\log (\zeta ^j + \zeta ^{-j}t)) + \sum _{ i=1}^5 \widehat{\Omega }^{i,6} d \log (\zeta ^i + \zeta ^{-i}t)=\) 
\(\sum _{1 \leqslant i < j \leqslant 6} \widehat{\Omega }^{i,j}d\log (\zeta ^i + \zeta ^{-i}t)+\sum _{1 \leqslant i < j \leqslant 6} \widehat{\Omega }^{i,j} d\log (\zeta ^j - \zeta ^{-j}t)) = -k^2 \operatorname{Id}\sum _{i=5}^6 d \log (\zeta ^i + \zeta ^{-i}t)\) 
where the last equality follows from (3) of propsition REF . This gives us that

\(\tilde{\psi }^*(\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
) =k ( 1 i < j 5 i,jd (i + -it - j - -jt) + k2 Idi=56 d (i + -it))

Now,
\(d \log (\zeta ^i + \zeta ^{-i}t - \zeta ^j - \zeta ^{-j}t)&={(\zeta ^{-i} - \zeta ^{-j}) dt \over (\zeta ^{i} - \zeta ^{j}) + (\zeta ^{-i} - \zeta ^{-j})t}\\&={dt \over {\zeta ^{i} - \zeta ^{j} \over \zeta ^{-i} - \zeta ^{-j}}+t} .\) 

This can be simplified by
\(-{\zeta ^{i} - \zeta ^{j} \over \zeta ^{-i} - \zeta ^{-j}} = {\zeta ^{i} - \zeta ^{j} \over \zeta ^{-j} - \zeta ^{-i}} {.} {\zeta ^ {i + j} \over \zeta ^ {i + j}} = \zeta ^{i +j},\) 
the result being
\(d \log (\zeta ^i + \zeta ^{-i}t - \zeta ^j - \zeta ^{-j}t)= {dt \over t - \zeta ^{i+j}}.\) 
Notice that
\(\zeta ^{i+j} = \zeta ^{k+l}\) 
if \(i+j=k+l (mod 5)\) .
Similarly,
\(d \log (\zeta ^i + \zeta ^{-i}t) &={ \zeta ^{-i} dt \over \zeta ^{i} + \zeta ^{-i}t}\\&= {dt \over t + \zeta ^{2i}}.\) 

This implies that

\(\tilde{\psi }^*(\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
) = k (1 i 5 Ai  dt t - i + k2 Id1i 5 dt t + 2i  )

which is projectively equivalent to ().
We define a connection in the trivial bundle \(H^0(^3, \mathcal {O}_{^3}(k)) \times \tilde{\mathchoice{\chi }{}{}{}}{\chi }\)\( as follows\begin{equation}\nabla ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{equation}\)\(\chi \)\(\chi \)  := d + \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)\(\chi \)\(\chi \) .

The connection () can be considered as a meromorphic connection on \( with logarithmic singularities at \) 5\(. The residue at each singularity \) i 5\( is \) Ai\(. See \cite {De} for an excellent introduction to meromorphic connections with logarithmic singularities.\)
We give an explicit expression of the parallel transport of () along paths in \(\tilde{\mathchoice{\chi }{}{}{}}{\chi }\)\( in terms of iterated integrals of 1-forms on \)\(\chi \)\(\chi \)\(. The 1-forms \) dt t - i\(, where \) i 5\(, give a basis for \) H1dR(\(\chi \)\(\chi \) , \(. Let \) [a,b] \(\chi \)\(\chi \)\( be a smooth map and for each \) i\( let\begin{equation}f_i(s_i) ds_i:=\gamma ^*\big ({dt \over t - \zeta ^i}\big ).\end{equation}The ordinary line integral of this 1-form is given by\begin{equation}\int _{\gamma } {dt \over t - \zeta ^i} = \int _a^b f_i(s_i)ds_i\end{equation}which is independent of the parametrization of \)\(. Now choose some r-tuple \) (i1, ..., ir) (5)r\( and consider the iterated integral\begin{equation}L_a(\zeta ^{i_1}, \dots , \zeta ^{i_r}|b): = \int _a^b f_{i_r}(s_{i_r}) \bigg (\int _a^{s_{i_r}} f_{i_{r-1}}(s_{i_{r-1}}) \dots \bigg (\int _a^{s_{i_2}} f_{i_1}(s_{i_1}) ds_{i_1}\bigg )\dots ds_{i_{r-1}}\bigg ) ds_{i_{r}}\end{equation}where \)\(f_{i_r}(s_{i_r}) ds_{i_r}=\gamma ^*\big ({dt \over t - \zeta ^{i_r}}\big ).\)\(This integral is independent of the choice of the parametrization of \)\( and depends on the homotopy class of \)\(. Such integrals, under the name of Hyperlogarithms, were introduced in \cite {Poi} and extensively studied in \cite {LD}.\) The iterated integrals \(L_0(\zeta ^{i_1}, \dots , \zeta ^{i_r}| 1)\)  appear in the study of the motivic fundamental group of \(*- \mu _n\)  (when specialized to n=5) carried out in [1]}. These iterated integrals also appear in the construction of the cyclotomic Drinfel'd associator carried out in [2]}.
Let us specialize to the path \([0, 1-\epsilon ] \subset \tilde{\mathchoice{\chi }{}{}{}}{\chi }\)\(, where \)\( is an arbitrarily small positive real number. In this case, for any \) (i1, ..., ir) (5)r\(, (\ref {hyperlogii}) is\begin{equation}L_0(\zeta ^{i_1}, \dots , \zeta ^{i_r}|1-\epsilon )= \int _0^{1-\epsilon } {1\over s_{i_r} - \zeta ^{i_r}} \bigg (\int _0^{s_{i_r}} {1\over s_{i_{r-1}} - \zeta ^{{i}_{r-1}}} \dots \bigg (\int _0^{s_{i_2}} {1\over s_{i_1} - \zeta ^{{i}_1}} ds_{i_1}\bigg )\dots ds_{i_{r-1}}\bigg ) ds_{i_{r}}.\end{equation}Define \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)  [0, 1 - ] End(H0(3, O3\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) ))\( by\begin{equation*}\Phi ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{equation*}\) (x) = Id+r=1kr    i1, ..., ir (5)r L0(i1, ..., ir| x)Ai1...Air          x [0, 1 - ],

where we consider \(L_0(\zeta ^{i_1}, \dots , \zeta ^{i_r}| x)\)  as a function in one variable: the upper limit of integration.
One easily proves that, see [3]} page 481 or [4]} page 164, that \(\Phi ^{\scriptscriptstyle (}k\) )\(\) (x)\( is convergent for all \) x [0, 1-] \( and\begin{equation}{d\,\Phi ^{\scriptscriptstyle (}k\scriptscriptstyle )}\end{equation}_{\epsilon }\over dx\)  = k 1 i 5 \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)   Ai x - i.

Since the connection () is given by the 1-form (), () implies that \(\Phi ^{\scriptscriptstyle (}k\) )\(\)\( gives the parallel transport of the connection (\ref {connectiontildechi}) along the path \) [0, 1 - ]\(.\) The integral () converges as \(\epsilon \rightarrow 0\)  if and only if \(\zeta ^{i_r} \ne 1\) , thus the integral \(L_0(\zeta ^{i_1}, \dots , \zeta ^{i_r}| 1)\)  is well defined if and only if \(\zeta ^{i_r} \ne 1\) . Let

\(\Phi ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
 := Id+r=1kr    i1, ..., ir (5)r ir1 L0(i1, ..., ir| 1)Ai1...Air      End(H0(3, O3\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) )).

The endomorphism (REF ) will play a crucial role in the monodromy computation carried out in the next section.
We now give a result on the asymptotic behavior of the parallel transport of () in arbitrarily small neighborhoods of the singularities \(\mu _5\) . Let \(D^*_i:=\lbrace  b_i \in  0 < |b_i| < \epsilon \rbrace \) , and consider the embedding

\(B_i \colon D^*_{i} \rightarrow \tilde{\mathchoice{\chi }{}{}{}}{\chi }\) 
,          bi i + bi .

An easy calculation shows

\(B^*_i (\omega ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
\(\chi \)\(\chi \) ) = k( Ai bi + 1 j 5 i j Aj    bi - (j - i)) dbi.

Let \(Y_i(b_i)\)  be an \(\operatorname{End}(H^0(^3, \mathcal {O}_{^3}(k))) \)  valued function over \(D^*_i\) . Then \(Y_i\)  is a solution of (REF ) if
\(Y_i^{\prime } = \hspace{-5.0pt}k\bigg ( {A_i \over b_i} + \sum _{\overset{1 \leqslant j \leqslant 5}{\ i \ne j}} {A_j\,\,\, \over b_i - (\zeta ^j - \zeta ^i)}\bigg ) Y_i.\) 
Proposition 3.6 
For all \(1 \leqslant i \leqslant 5\) , there exists a unique \(Y_i\)  satisfying (REF ) such that
\(Y_i (b_i) = Q_i(b_i) \,b_i^{\hspace{-5.0pt}kA_i}\) 
where \(Q_i(b_i) = \sum _{r\geqslant 0} q^{\scriptscriptstyle (}i\) )\(\) r bir\(, with \) qr\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\)  End(H0(3, O3(k)))\(, and \) Qi(0)=q0\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\)  = Id\(.\)
Remark 3.7 For the expression \(b_i^{\hspace{-5.0pt}kA_i}\)  to make sense on \(D^*_i\) , we must choose a branch cut for the logarithm. We make the choice of the positive real axis in \(D^*_i\)  as the branch cut for the logarithm.

By the theory of ordinary differential equations, see [5]}, there exists a solution \(Y_i\)  satisfying (REF ). We show that there exits a family \(q_r^{\scriptscriptstyle (}i\) )\(\)\( satisfying the above stated requirements such that \) Yi\( is of the form (\ref {localsolution}). We write out\begin{equation}Y_i^{\prime }(b_i)= \bigg (Q_i^{\prime }(b_i) + \hspace{-5.0pt}k{Q_i(b_i)A_i \over b_i}\bigg ) b_i^{\hspace{-5.0pt}kA_i}=\hspace{-5.0pt}k\bigg ({A_i \over b_i} + \sum _{\overset{1 \leqslant j \leqslant 5}{\ i \ne j}} {A_j\,\,\, \over b_i - (\zeta ^j - \zeta ^i)}\bigg ) Q_i(b_i) \,b_i^{\hspace{-5.0pt}kA_i}.\end{equation}This can be rewritten as\begin{equation}b_i Q_i^{\prime }(b_i) - \hspace{-5.0pt}k[A_i, Q_i(b_i)]= -\hspace{-5.0pt}k\,\,\sum _{\overset{1 \leqslant j \leqslant 5}{\ i \ne j}} A_j {b_i \,Q_i(b_i) \over (\zeta ^j - \zeta ^i)- b_i}.\end{equation}Expanding the above in powers of \) bi\(, we get \) [Ai, q0\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) ]=0\(, and for \) r >0\(\begin{equation}r q^{\scriptscriptstyle (}i\scriptscriptstyle )\end{equation}\) r - k[Ai, q\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) r] = - kl=1r (1 j 5 i j Aj(j - i)l )q\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) l-1

The above equations, for every \(r\) , have a solution. Indeed, if we take \(q^{\scriptscriptstyle (}i\) )\(\) 0=Id\(, then \) q\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) r\( is uniquely determined by \) q\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) 0, ..., q\(\scriptscriptstyle (\) i\(\scriptscriptstyle )\) r-1\( due to the fact that the operator \) r Id- k  ad(Ai)\( is invertible with inverse equal to\begin{equation}{1 \over r} \sum _{n \geqslant 0} {\hspace{-5.0pt}k^n \over r^n} \operatorname{ad}(A_i)^n.\end{equation}The convergence of \) Qi(bi)\( results from the general fact that a formal solution of a regular singular equation is necessarily convergent, see \cite {Kass} and \cite {Was}.\)
Generators of the (orbifold) fundamental group
The curve \(\mathchoice{\chi }{}{}{}\)\(\chi \)\( is defined as the quotient \)\(\chi \)\(\chi \) /D\(, where \) D\( was generated by the automorphisms \) R, I \(\chi \)\(\chi \)  \(\chi \)\(\chi \)\(. The rotation \) R\( is of order five and has two fixed points \) 0\( and \)\(, the involution \) I\( has the fixed point \) -1\(, it follows that \)\(\chi \)\(\chi \)\( is topologically a sphere with two orbifold points \) a := \(\chi \)\(\chi \) (0)\(, \) b:= \(\chi \)\(\chi \) (-1)\( and one puncture. The stabilizer group of the orbifold point \) a\( is \) R\(, and the stabilizer group of the orbifold point \) b\( is \) I\(.\)
We choose explicit paths based at the order five orbifold point of \(\mathchoice{\chi }{}{}{}\)\(\chi \)\( such that they generate the group \) 1orb(\(\chi \)\(\chi \) , a)\(. Let \) 0 \(\chi \)\(\chi \)\( be the path starting from \) 0\( and running along the real axis to \) 1-\(, where \)\( is an arbitrarily small positive real number. Let \) 1\( be the semi-circle starting from \) 1- \( moving around \) 1\( in an anti clockwise direction until it reaches \) (1 - )-1\(. We will also need \) 0-1\( which is the path running from \) 1-\( to \) 0\( along the real axis.\) The end points of the semi-circle \(\tilde{\gamma }_1\)  are identified with each other under the involution \(I\) , which implies that the projection 
\(\pi _{\tilde{\mathchoice{\chi }{}{}{}}{\chi }}{\chi }{\chi }\) 
(1)=:1 \(\chi \)\(\chi \)
\( is a homotopically non-trivial closed loop. Consider the projection of the path \) 
0\(, \)\(\pi _{\tilde{\mathchoice{\chi }{}{}{}}{\chi }}{\chi }{\chi }\) (0)=: 0 \(\chi \)\(\chi \)
\( then the composition\begin{equation}\gamma _0^{-1}\cdot \gamma _1\cdot \gamma _0=: \gamma \subset \mathchoice{\chi }{}{}{}\end{equation}{\chi }\) 

is a closed, connected, homotopically non-trivial loop with the property that it starts and ends at the order five orbifold point \(a \in \mathchoice{\chi }{}{}{}\)\(\chi \)\(.\)
The following is part of proposition 4.11 from [6]}.
Proposition 4.1 
The loop \(\gamma \)  and the orbifold stabilizer group \(\langle R \rangle \)  together generate \(\pi _1^{orb}(\mathchoice{\chi }{}{}{}\)\(\chi \) , a)\(, and under the isomorphism \) 1orb(\(\chi \)\(\chi \) , a) (2, 5, )\(R is identified with \) U:=ST\( and \)\( is identified with \) T\(.\) 
Let

\(H : \pi _1^{orb}(\mathchoice{\chi }{}{}{}\) 
\(\chi \) , a) H1((a), Z)Sp(4, Z)

be the monodromy representation provided by the Gauss-Manin connection in the bundle of the first de Rahm cohomology groups of the genus two surfaces parametrized by \(\mathchoice{\chi }{}{}{}\)\(\chi \)\(.\begin{lemma}The elements \langle R\rangle  and \gamma  map to M_0 and M_1 respectively where\begin{equation}M_{0}:=\begin{bmatrix}0 & 0 & -1 & 1\\0& 0 & 0 & -1\\1 & 0 & -1 & 0\\1 & 1 & -1 & 0\end{bmatrix} \quad M_{1}:=\begin{bmatrix}0 & 1& 0 & 0\\1&0 & 0 & 0\\1 & 1 & 0 & -1\\1& 1 & -1 & 0\end{bmatrix}.\end{equation}under the homomorphism H.\end{lemma}{\begin{xmlelement*}{proof}Let X_{\sqrt{5}} be the Hilbert modular surface for totally real quadratic field \mathbb {Q}(\sqrt{5}) considered in \cite {CMZ}. The surface X_{\sqrt{5}} is the moduli space of abelian surfaces which admit real multiplication by the field \mathbb {Q}(\sqrt{5}). In \cite {CMZ} the image of Gauss-Mannin connection H^{\prime }: \pi ^{orb}_1(X_{\sqrt{5}}, *) \rightarrow H_1( A_*, \mathbb {Z})\cong \mathrm {Sp}(4, \mathbb {Z}) is described explicitly. In \cite {Mc} it is shown that the Teichmüller curve \mathchoice{\chi }{}{}{}\end{xmlelement*}{\chi }}{\chi }{\chi }\hookrightarrow X_{\sqrt{5}}\)  embeds in the hilbert modular surface; In particular an embedding \(\pi _1^{orb}(\mathchoice{\chi }{}{}{}\)\(\chi \) , a) 1orb(X5, *)\( is provided. We simply look at the image of \) <R>\( and \)\( under the composition of this embedding with \) H'\(.\)
Let us choose \(\phi (a) \in \mathcal {M}_2\)  as the base point for the orbifold fundamental group of \(\mathcal {M}_2\) . Then \(\phi (\gamma ) \in \pi _1^{orb}(\mathcal {M}_2, \phi (a))\) . In fact, see [7]} or [6]}, the loop \(\phi (\gamma )\)  corresponds to a Dehn twist along two non-separating simple closed loops on a genus two surface, and \(R\)  corresponds to an order five diffeomorphism.
For each integer \(k >0\)  we have the projective vector bundle with the flat connection. The monodromy of this flat connection provides us with a group homomorphism

\(\rho ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
 1orb(M2, (a)) End(PV\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (a))

where the fiber \( \operatorname{End}(\mathbb {P}\overline{\mathbb {V}}^{\scriptscriptstyle (}k\) )\(\) (a))\( can be identified with \) End(PH0(3, O3(k)))\(. In the next section, we compute the image of \) ()\( and \) R\( under the morphism (\ref {monodromyrep}), and prove theorem \ref {maintheorem}.\)
Computing Monodromy
In order to compute the image of \(\phi (\gamma ) \)  and \(R\)  under (REF ) we need two connected paths, \(\tilde{\gamma }\)  and \(p_R\) , based at some point \(\bf {z}\in \mathcal {C}\)  such that \(\pi _{\mathcal {M}_2}(\)\(\bf {z}\)\()=\phi (a)\)  and \(\pi _{\mathcal {M}_2}(\tilde{\gamma })=\phi (\gamma )\) , \(\pi _{\mathcal {M}_2}(p_R)=\phi (a)\) .
We have the following well known isomorphism, see [9]} page 188,
\(\pi _1^{orb}(\mathcal {M}_2, *)/(\mathbb {Z}/2) \cong SB_6\) 
where \(\mathbb {Z}/2=\langle \sigma \rangle \)  and \(\sigma \)  is the hyperelliptic involution.
Consider the following map
\(p^{\prime }_I \colon [1, (1-\epsilon )^{-1}] \rightarrow \mathbb {P}\mathrm {SL}(2,  \quad s \mapsto \begin{bmatrix} s^{1\over 2} & 0\\ 0 & s^{-{1\over 2}} \end{bmatrix}.\) 
Let \(p_I \subset \mathcal {C}\)  be the path obtained by applying the family of Möbius transformations \(p^{\prime }_I\)  to the point \(\tilde{\psi }(1-\epsilon ) \in \mathcal {C}\) .
Proposition 4.2 The following
\(\tilde{\gamma }:=\nu ^{\prime }(\tilde{\psi }((\tilde{\gamma }_0)^{-1})))\cdot p_I \cdot \tilde{\psi }(\tilde{\gamma }_1) \cdot \tilde{\psi }(\tilde{\gamma }_0) \subset \mathcal {C}\) 
is a connected path with initial point \(\tilde{\psi }(0)\) , moreover \(\pi _{\mathcal {M}_2}(\tilde{\gamma }) = \phi (\gamma )\) .

Let us first analyze the image \(\tilde{\psi }(\tilde{\gamma }_0)\) . The initial point of the path \(\tilde{\psi }(\tilde{\gamma }_0)\)  is \(\tilde{\psi }(0)\)  which is shown in Figure REF . Here the six points form a symmetric configuration by taking the positions at the fifth roots of unity and zero in the specified order.
<FIGURE>In Figure  the dashed lines show the trajectory of the six points as one travels along \(\tilde{\psi }(\tilde{\gamma }_0)\) . At the end point of \(\tilde{\psi }(\tilde{\gamma }_0)\) , the points \(z_1\)  and \(z_4\)  are a small distance (depends on \(\epsilon \) ) away from the positive real number \({1 \over 2 \mathrm {Re} (\zeta )}\)  and a quick calculation shows that at the end points \(z_1\)  and \(z_4\)  are conjugate to each other, and both have their real parts bigger than \({1 \over 2 \mathrm {Re} (\zeta )}\) . Moreover, \(z_1\)  lies in lower half plane of \( and \) z4\( lies in the upper half plane. This local picture at \) 1 2 Re ()\( is shown in Figure \ref {Fig3}. A similar story holds for the pair \) z2\( and \) z3\( around the negative real number \) 1 2 Re (2)\(. The point \) z5\( moves to \) 12\( and \) z6\( remains fixed at zero. Notice also that the end point of \) (0)\( is the initial point of \) (1)\(.\begin{figure}[ht]\includegraphics [height=3in,width=5in,angle=0]{loops}\caption {}\end{figure}We now look at the configuration at the end point of \) (1)\(. Recall that \) 1\( was a semi-circle traversed anti-clockwise around one from \) (1- )\( to \) (1-)-1\(. At the end point of \) (1)\(, the points \) z1\( and \) z4\( are still conjugate to each other but both have real parts smaller than \) 1 2Re()\( and \) z1\( lies in the upper half plane and \) z4\( lies in the lower half plane. In effect, both \) z1\( and \) z4\( move in an anti-clockwise semi-circle around \) 12Re()\(. This is also shown in figure \ref {Fig3}. A similar story holds for the pair \) z2\( and \) z3\( around the negative real number \) 1 2Re(2)\(. The point \) z5\( moves to \) 12\( and \) z6\( remains fixed at zero. Notice also that the end point of \) (1)\( is not the initial point of \) (0)-1\(. However, initial point of \) pI\( is the end point of \) (1)\( and it follows from\begin{equation*}\tilde{\psi }(1-\epsilon )= \nu ^{\prime } (1-\epsilon )^{-1} (\tilde{\psi }(1-\epsilon )^{-1})\end{equation*}that the configuration at the end point of \) pI\( differs from the configuration at the initial point of \) (0)-1\( only by the permutation \) '\(.\) The trajectory of \(z_4\)  and \(z_1\)  along the path \(p_I\)  is shown in red in figure REF . A similar story holds for \(z_2\)  and \(z_3\) . The point \(z_5\)  moves a little to the right and \(z_6\)  is fixed at zero.
<FIGURE>Since the end point of \(p_I\)  differs from the initial point of \((\tilde{\psi }(\tilde{\gamma }_0)^{-1})\)  by the permutation \(\nu ^{\prime }\) , it follows that \(\nu ^{\prime }(\tilde{\psi }(\tilde{\gamma }_0)^{-1}))\cdot p_I\)  is a connected path with initial point \(\tilde{\psi }((1-\epsilon )^{-1})\)  and end point \(\nu ^{\prime }(\tilde{\psi }(0)\) .
Since the initial point of \(\nu ^{\prime }(\tilde{\psi }(\tilde{\gamma }_0)^{-1}))\cdot p_I\)  is the end point of \(\tilde{\psi }(\tilde{\gamma }_1)\cdot \tilde{\psi }(\tilde{\gamma }_0)\)  it follows that \(\tilde{\gamma }\)  is a connected path in \(\mathcal {C}\)  with initial point \(\tilde{\psi }(0)\)  and end point \(\nu ^{\prime }(\tilde{\psi }(0))\) .
Since \(p_I\)  is entirely contained in the \(\mathbb {P}\mathrm {SL}(2, \)  orbit, it follows that \(\pi _{\mathcal {M}_2} (p_I)\)  is a constant loop in \(\mathcal {M}_2\) . This fact combined with lemma REF  implies
\(\pi _{\mathcal {M}_2}(\tilde{\gamma })=\phi (\gamma ).\) 
Consider the following path in \(\mathbb {P}\mathrm {SL}(2, \)
\(p_R \colon [0, {1 \over 5}] \rightarrow \mathbb {P}\mathrm {SL}(2, , \quad \quad s \mapsto \begin{bmatrix} e^{- \pi i s} & 0 \\ 0 & e^{ \pi i s} \end{bmatrix}\,\,\, s\in [0, {1 \over 5}].\) 
Denote also by \(p_R\)  the path in \(\mathcal {C}\)  given by the action of \(p_R\)  on the point \(\psi (0)\) .
Lemma 4.3 
The path \(p_R \subset \mathcal {C}_{\infty }\)  is connected with initial point \(\widetilde{\psi }(0)\) , moreover \(\pi _{\mathcal {M}_2}(p_R)= \phi (a)\) .

At both the initial and the end point of this path the first five points sit at the fifth roots of unity (sixth at zero) but with different ordering. The ordering differs by \(\nu ^{-1}\) . Since the end points of \(p_R\)  are related by an element in \(S_6\) , \(P_{\infty }(p_R) \subset \overline{\mathcal {C}}\)  is a closed loop. Moreover, since \(p_R\)  is entirely contained in the \(\mathbb {P}\mathrm {SL}(2, \)  orbit, \((\overline{P}_{\infty }\cdot P_{\infty })(p_R)\)  is a constant path in \(\mathcal {M}_2\) , namely \(\phi (a)\) .
<FIGURE>We now prove theorem REF  from the introduction.
(Theorem REF )
We compute the parallel transport of \(\nabla ^{\scriptscriptstyle (}k\) )\(\) C\( along \)\( and \) pR\(. Recall from (\ref {tildegamma})\begin{equation}\tilde{\gamma }:=\nu ^{\prime }(\psi ((\tilde{\gamma }_0)^{-1})))\cdot p_I \cdot \psi (\tilde{\gamma }_1) \cdot \psi (\tilde{\gamma }_0) \subset \mathcal {C}\end{equation}and \) pR\( is given in lemma \ref {60}.\)
We first observe the following
Lemma 4.4 The parallel transport of \(\nabla ^{\scriptscriptstyle (}k\) )\(\) C\( along the paths \) pI\( and \) pR\( is projectively trivial.\)
Recall that \(\nabla ^{\scriptscriptstyle (}k\) )\(\) C:= d + \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)\( and \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)\( is projectively invariant under the \) PSL(2, \( on \) C\(. Lemma follows since both \) pR\( and \) pI\( are contained in the orbit of the dilation subgroup of \) PSL(2, \(.\)
In order to compute the parallel transport along \(\tilde{\gamma }\)  we must compute the parallel transport along all the paths separately appearing in the expression above. We compute the parallel transport along \(\widetilde{\psi }(\tilde{\gamma }_1)\) . The semi circle \(\tilde{\gamma _1} \subset \tilde{\mathchoice{\chi }{}{}{}}{\chi }\)\( is contained in the image of \) B1\(, defined in (\ref {discs}), which is a punctured disk of radius \)\( based at \) 1 \(\chi \)\(\chi \)\(.\)
Proposition 4.5 
The parallel transport along \((\widetilde{\psi }\circ B_1)^{*}(\psi (\tilde{\gamma }_1))\)  with respect to \((\widetilde{\psi }\circ B_1) ^{*}\nabla ^{\scriptscriptstyle (}k\) )\(\)\( is\begin{equation}Q_1(-i\epsilon )\cdot \exp \big (-\pi i \hspace{-5.0pt}kA_1 \big )\cdot Q_1(i \epsilon ),\end{equation}where \) Q1(b1)\( is the power series from proposition \ref {localsolutionproposition}.\)
Remark 4.6 We assume the \(\epsilon \)  in the definition of \(B_1\)  to be bigger than the \(\epsilon \)  in the definition of \(\tilde{\gamma }_1\) . The \(\epsilon \)  appearing in the proposition refers to the \(\epsilon \)  for \(\tilde{\gamma }_1\) .

From proposition REF  we know that the parallel transport of
\((\widetilde{\psi }\circ B_1 )^{*}\nabla ^{\scriptscriptstyle (}k\) )\(\)\( is given by a solution of the form \) Y1(b1)\( shown in (\ref {localsolution}). The pull back \) (B1)*((1))\( is a semi circle in \) D1\( traveling in an anti-clockwise direction from \) i\( to \) -i \(, avoiding the positive real axis in \) D1\( (which was our choice for the branch cut of the logarithm on \) D1\(). Thus the parallel transport is given by{\begin{@align*}{1}{-1}Y_1(-i\epsilon )(Y_1(i\epsilon ))^{-1} &= Q_1(-i\epsilon ) (- i \epsilon )^{\hspace{-5.0pt}kA_1} (i\epsilon ^{-\hspace{-5.0pt}kA_1})Q_1(i\epsilon )^{-1} \\& = Q_1(-i\epsilon ) \exp (log(-i\epsilon ) \hspace{-5.0pt}kA_1)\exp (-log (i\epsilon ) \hspace{-5.0pt}kA_1) Q_1(i\epsilon )^{-1}\\&= Q_1(-i\epsilon ) \exp ((log(i\epsilon ) + \pi i)\hspace{-5.0pt}kA_1) \exp (-log (i\epsilon ) \hspace{-5.0pt}kA_1) Q_1(i\epsilon )^{-1}\\&=Q_1(-i\epsilon ) \exp (\pi i \hspace{-5.0pt}kA_1) Q_1(i\epsilon )^{-1},\end{@align*}}where the first equality is from (\ref {localsolution}) and third equality follows from our choice of the branch cut for the logarithm.\)
For the parallel transport of \(\nabla ^{\scriptscriptstyle (}k\) )\(\) C\( along \) (0)\(, where recall that \) 0 \(\chi \)\(\chi \)\( is the interval \) [0, 1 - ]\(, we take the pull back \) *(\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) )\(, which is given in (\ref {psipullback}), restrict it to the interval \) [0, 1 - ]\(, and compute the monodromy there. This monodromy was computed in section \ref {iteratedintegralsection} and is given by \)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1- )\( c.f. (\ref {30}).\)
Now, it follows that under the map (REF ) we have that

\(\rho ^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
(R) = M\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) 0

since parallel transport along \(p_R\)  is projectively trivial. Here \(M_0^{\scriptscriptstyle (}k\) )\(\)\( denotes the \) k\(-th symmetric power of the endomorphism \) M0\( from lemma \ref {m0m1}.Likewise we have that{\begin{@align*}{1}{-1}\rho ^{\scriptscriptstyle (}k\scriptscriptstyle )\end{@align*}}(\phi (\gamma )) &=& M^{\scriptscriptstyle (}k\) )\(\) 1 ' (\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - ))-1 Q1(-i) exp(-i kA1 )(Q1(i ))-1\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - )
= (\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - ))-1 M1\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (Q1(-i) exp(-i kA1 )Q1(i )-1)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - )

where we used the fact that \(M_1^{\scriptscriptstyle (}k\) )\(\)\( commutes with the parallel transport, see \cite {GJ}.This commutation also had the effect ofremoving \) '\( from\) '(\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - ))-1)\( because \) M1\( maps to \) '\(under the map \) A(G) S6\(.\)
In [2]}, see also [3]} and [4]}, it is shown that
\(&\lim _{\epsilon \rightarrow 0}&\Big ((\Phi _{\epsilon }^{\scriptscriptstyle (}k\scriptscriptstyle )\) 
(1 - ))-1 M1\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (Q1(-i) exp(-i kA1 )Q1(i )-1)\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (1 - ))
= (\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) )-1((M\(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) 1)-1exp( -i kAi)) \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\)
= \(\scriptscriptstyle (\) k\(\scriptscriptstyle )\) (())

where \(\Phi ^{\scriptscriptstyle (}k\) )\(\)\( is defined in (\ref {911}).\) Replacing \(R\)  with \(U\)  and \(\phi (\gamma )\)  with \(T\)  under the isomorphism \(\pi _1^{orb}(\mathchoice{\chi }{}{}{}\)\(\chi \) , a) (2, 5, )\( we get the statement of theorem \ref {maintheorem}.\)
